I failed to understand the definition of holomorphic $1$-form on Riemann surfaces. Can one explain it here? I saw two definitions in the books of Miranda and Farkas-Kra.
Definition 1.: Suppose that $w_1=f(z)dz$ is a holomorphic $1$-form in the coordinate $z$, defined on an open set $V_1$. Also suppose that $w_2=g(w)dw$ be a holomorphic $1$-form in the coordinate $w$, defined on an open set $V_2$. Let $T$ be a holomorphic mapping from open set $V_2$ to $V_1$. We say that $w_1$ transforms to $w_2$ under $T$ if
(1) $g(w)=f(T(w))T'(w)$.
Let $X$ be a Riemann surface. A holomorphic $1$-form on $X$ is a collection of holomorphic $1$-forms $\{w_{\phi}\}$, one for each chart $\phi\colon U\rightarrow V $ in the co-ordinate of the target $V$, such that if two charts $\phi_i\colon U_i\rightarrow V_i$ (for $i=1,2$) have overlapping domains then the associated $1$-form $w_{\phi_1}$ transforms to $w_{\phi_2}$ under the change of coordinate mapping $T=\phi_1 \circ \phi_{2}^{-1}$.
Question 1: Can one explain the meaning of (1)? Can we say anything about it in terms of commutative diagram?
Definition 2. Let $M$ be a Remann surface. A $1$-form $w$ on $M$ is an (ordered) assignment of two continuous functions $f$ and $g$ to each local coordinate $z(=x+iy)$ on $M$ such that
(2) $ fdx + gdy$
is invariant under coordinate changes; that is, if $\tilde{z}$ is another local coordinate on $M$ and the domain of $\tilde{z}$ intersects non-trivially with the domain of $z$, and if $w$ assigns the functions $\tilde{f},\tilde{g}$ to $\tilde{z}$, then
(3) $ \tilde{f}(\tilde{z})=\frac{\partial x}{\partial \tilde{x}} f(z(\tilde{z})) + \frac{\partial y}{\partial \tilde{x}} g(z(\tilde{z})) $
(4) $ \tilde{g}(\tilde{z})=\frac{\partial x}{\partial \tilde{y}} f(z(\tilde{z})) + \frac{\partial y}{\partial \tilde{y}} g(z(\tilde{z})) $
Question 2 What is the meaning of these equations? How did they come (or how we imposed these conditions for the definition of holomorphic $1$-form? Can we say anything about them in terms of commutative diagrams.
Best Answer
I will assume that you know how to handle smooth differential forms on smooth real manifolds, this is a necessary prerequisite for understanding complex and holomorphic differential forms on complex manifolds.
(I won't mention commutative diagrams because I don't know how to draw these here, because I don't know which ones would be useful and because I'm confident that you'll come up with the ones you find useful yourself after you have understood the definition in the most simple setting.)
Suggestion: Let's look at the most simple situation, $\mathbb{C}$ as both a one dimensional complex manifold and as a two dimensional smooth real manifold $\mathbb{R}^2$.
On $\mathbb{R}^2$ we have a global chart, the cartesian coordinates, with coordinates denoted by $(x, y)$. In every point we have a basis of the tangential space $(\partial_x, \partial_y)$ and the dual basis of differential forms in the cotangential space $(d x, d y)$. Every (smooth) differential form can therefore be written (in this chart) as $$ d w = f_1(x, y) d x + g_1(x, y) d y $$ where we say that $d w$ is smooth if $f$ and $g$ are.
Now we can make a coordinate change in $\mathbb{C} = \mathbb{R}^2$ to the new coordinates $$ z = x + i y $$ and $$ \bar{z} = x - i y $$ This is an example of a biholomorphic mapping of an open set $V_2 = \mathbb{C}$ to $V_1 = \mathbb{C}$, which is therefore a change of coordinates on $\mathbb{C}$ (both as a real smooth and as a complex manifold). Now we can express every differential form $w$ again in the new coordinates: $$ d w = f_2(z, \bar{z}) d z + g_2(z, \bar{z}) d \bar{z} $$ The $f_2, g_2$ are related to $f_1, g_1$ (how?).
Motivation for the definition of "holomorphic differential form":
The differential $d f$ of a smooth function $f$ is a real smooth differential form. Similarily, one would like to define "holomorphic differential form" in a way such that the differential $d f$ of a holomorphic function $f$ is a holomorphic differential form. Since $f$ is holomorphic iff it depends on the coordinate $z$ only, one defines a holomorphic differential form to be a differential form that does depend on the coordinate $z$ only, that is $d w$ is holomorphic iff it can be written as $$ d w = g(z) d z $$ with a holomorphic function $g(z)$. This definition makes explicitly use of the global coordinate chart of $\mathbb{C}$ that we defined above. So, in order to show that "holomorphic differential form" is well defined on $\mathbb{C}$ seen as a complex manifold, we need to show that coordinate changes on $\mathbb{C}$ do not map holomorphic differential forms to non-holomorphic or vice versa.
A coordinate change in our example would be any biholomorphic map of an open subset of $\mathbb{C}$ to another open subset of $\mathbb{C}$. Suggestion: Pick an easy example, write $d w$ in real coordinates and do a coordinate transform.
After that, have another look at the two definitions you cited.